Biased Linearity Testing in the 1% Regime

2025-02-04

AI Summary

This work investigates the feasibility of linearity testing on the $p$-biased hypercube $({0,1}^n, mu_p^{otimes n})$ with 1% tolerance. For $k$-query linear tests $mathrm{Lin}( u)$, we establish the exact threshold for test efficacy: such a test reliably detects near-linear functions—i.e., functions passing with probability $geq 1/2 + varepsilon$ must agree with some linear function on a $mu_p^{otimes n}$-measure $geq 1/2 + delta$, where $delta$ depends only on $varepsilon$—if and only if $k geq 1 + 1/p$. Our key conceptual contribution is the equivalence between test efficacy and pairwise independence of the query distribution $ u$, yielding a tight bidirectional characterization linking theoretical thresholds to structural conditions. Methodologically, we integrate tools from probabilistic Boolean analysis, $p$-biased measure theory, high-dimensional distribution construction, and an extended analysis of the Bhangale–Khot–Minzer framework.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study linearity testing over the $p$-biased hypercube $({0,1}^n, mu_p^{otimes n})$ in the 1% regime. For a distribution $ u$ supported over ${xin {0,1}^k:sum_{i=1}^k x_i=0 ext{ (mod 2)} }$, with marginal distribution $mu_p$ in each coordinate, the corresponding $k$-query linearity test $ ext{Lin}( u)$ proceeds as follows: Given query access to a function $f:{0,1}^n o {-1,1}$, sample $(x_1,dots,x_k)sim u^{otimes n}$, query $f$ on $x_1,dots,x_k$, and accept if and only if $prod_{iin [k]}f(x_i)=1$. Building on the work of Bhangale, Khot, and Minzer (STOC '23), we show, for $0<p leq frac{1}{2}$, that if $k geq 1 + frac{1}{p}$, then there exists a distribution $ u$ such that the test $ ext{Lin}( u)$ works in the 1% regime; that is, any function $f:{0,1}^n o {-1,1}$ passing the test $ ext{Lin}( u)$ with probability $geq frac{1}{2}+epsilon$, for some constant $epsilon>0$, satisfies $Pr_{xsim mu_p^{otimes n}}[f(x)=g(x)] geq frac{1}{2}+delta$, for some linear function $g$, and a constant $delta = delta(epsilon)>0$. Conversely, we show that if $k<1+frac{1}{p}$, then no such test $ ext{Lin}( u)$ works in the 1% regime. Our key observation is that the linearity test $ ext{Lin}( u)$ works if and only if the distribution $ u$ satisfies a certain pairwise independence property.

Problem

• Studies linearity testing on p-biased hypercubes.
• Analyzes k-query linearity test conditions.
• Determines effectiveness of linearity tests in 1% regime.

Innovation

• Uses p-biased hypercube for testing
• Applies k-query linearity test
• Requires pairwise independence property